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%%文档的题目、作者与日期
%\author{王立庆（2020级数学与应用数学1班） }
\author{学号 \underline{\hspace{4cm}} \hspace{1cm} 姓名 \underline{\hspace{4cm}} }
\title{常微分方程期中考试}
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\date{2023 年 11 月 23 日}
%\date{March 9, 2021}

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\begin{document}

\maketitle

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\begin{enumerate}

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\item  %第1题
判断下列方程是否为恰当方程，并对恰当方程求解：$$(y^2+2)\cos(x)dx + 2y[\sin(x)+\cos(y)]dy=0. $$

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\item  %第2题
求解下述微分方程的初值问题：$$\sqrt{1+x^2}\frac{dy}{dx}=xy^3,\,\, y(0)=2. $$

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\item  %第3题
使用变量代换 $u=\sin(y)$, 求解微分方程：$$\frac{dy}{dx}=\frac{\cos(x)}{\cos(y)} + \tan(y). $$

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\item  %第4题
考虑变量代换 $u=\frac{y'}{y}$, 将微分方程 $y''+ay'+by=0$ 化成关于 $u(x)$ 的微分方程。%，并求解该方程。

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\item  %第5题
考虑初值问题 $\frac{dy}{dx} = F(x,y),\, y(0)=0$, 其中函数 $F(x,y)$ 在矩形区域 $[0,1]\times (-\infty,\infty)$ 分区域定义。 
\begin{eqnarray*}
F(x,y) = \left\{\begin{array}{ll}
0, & x=0, -\infty<y<\infty, \\
2x, & 0<x\le 1, -\infty<y<0, \\
2x-\frac{4y}{x}, & 0<x\le 1, 0\le y<x^2, \\
-2x, & 0<x\le 1, x^2\le y<\infty.
\end{array}\right.
\end{eqnarray*}
计算该初值问题的皮卡序列的前四个函数。

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\item  %第6题
记 $p=\frac{dy}{dx}$, 求微分方程 $xp^3-yp^2-1=0$ 的所有解。

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\item  %第7题
\begin{enumerate}
\item  将微分方程 $\frac{d^2x}{dt^2} = -\sin(x)$ 写成一阶微分方程组的形式。
\item  设 $v=\frac{dx}{dt}$, 将微分方程 $\frac{d^2x}{dt^2} = -\sin(x)$ 化为关于 $v$ 与 $x$ 的微分方程，并求解该微分方程。
\end{enumerate}

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\item  %第8题
求经过点 $(1,1)$ 的曲线，使其与双曲线族 $x^2-4y^2=C$ 都垂直。
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\end{enumerate}


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\end{document}

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